Taylor 8.1-8.3 (today) and 8.3-8.5 (Wednesday).
We will now look at the general problem of two bodies interacting by a central force. This is the problem that Newton developped the calculus and his mechanics to solve, specifically for the orbits of planets and comets about the Sun. Many similar problems arise in Physics: the orbits of binary stars, including neutron stars and black holes; the interaction of two atoms in a diatomic molecule such as H2, N2, O2, or CO; and the behavior of an electron bound to a nucleus. The last two are properly treated only with quantum mechanics, but by studying the problem using classical mechanics we will (1) learn some important concepts about the problem that are applicable in quantum mechanics, and (2) understand how classical mechanics fails in such situations. The problem of orbiting neutron stars or black holes also is beyond the validity of classical mechanics, and must be treated with general relativity.
Taylor uses the language of Lagrange mechanics to arrive at the equations of motion in this chapter. I will derive the equations of motion using Newtonian mechanics.
Recall that in chapter 4 we derived that the motion of a system of particles can be separated into the problem of the motion of the center of mass of the system subjected to the sum of all external forces and the internal motion of the system relative to the center of mass. Consider the two objects to be point-like, that is, small enough that we never have to consider their size. Let their positions be r1 and r2, and their masses m1 and m2. A two body central force is automatically conservative. Consider the two objects to be point-like, that is, small enough that we never have to consider their size. Let their positions be given by the vectors r1 and r2, then the potential is U(r1, r2) = U(|r1-r2|) = U(r), where r is just the separation of the two objects. The corresponding forces are F12 = -∇1U and F21 = -∇2U = ∇1U = -F12. Assuming there are no other forces, internal or external, the equations of motion are
The center of mass is given by R = (m1r1 + m2r2) / (m1 + m2). Calling M = m1 + m2, we can write this as MR = m1r1 + m2r2. The equation of motion for the center of mass is
That is, the total external forces are zero, therefore the center of mass moves at constant velocity.
The internal motion is the topic of interest. This can be investigated using relative coordinates r'1 = r1 - R and r'2 = r2 - R. In terms of the relative coordinates, the equations of motion become